Linear equations in one variable are fundamental mathematical tools used to solve problems involving a single unknown quantity. These equations have the general form a x plus b equals zero, where a and b are known numbers and x is the variable we need to find. Let's look at a simple example: three x plus five equals fourteen.
Linear equations in one variable can solve many types of real-world problems. These include finding unknown numbers, solving age-related problems, calculating distance and speed, work and time problems, money calculations, and percentage problems. Let's work through an age problem example. John is three times older than his sister, and in five years he will be twice as old as she will be then. How old is John now?
Distance and speed problems are common applications of linear equations. These problems use the fundamental relationship that distance equals speed times time. Let's solve an example: A car travels one hundred twenty kilometers in two hours. If it increases its speed by ten kilometers per hour, how long will the same trip take? We first find the original speed, then calculate the new speed, and finally determine the new time required.
Money and percentage problems are practical applications of linear equations that we encounter in daily life. These include cost calculations, profit and loss scenarios, interest calculations, and discount problems. Let's solve a discount problem: A store offers a twenty percent discount, and you save fifteen dollars. What was the original price? We set up the equation using the relationship that discount amount equals original price times the discount rate.
To summarize what we have learned: Linear equations in one variable are powerful tools for solving problems with a single unknown. They apply to many real-world situations including age problems, distance and speed calculations, money and percentage problems. The key to success is identifying the relationships between quantities and setting up the correct equation. Always remember to check your answer to ensure it makes sense in the context of the original problem.